It is well-known that \(L\)  is a positive semidefinite matrix. More about the properties
of the Laplacian matrices can be found in [1]}, [2]}. In our proofs we use the fact that the largest eigenvalue \(\lambda _{max}\)  of \(L\)  satisfies \(\lambda _{max}\le n\) , that is, \(\frac{\lambda _{\max }}{n}\le 1.\)
